How to bevel correct way
You want to make sure that when you are using bevels you don't want any ngons  on your model. These are faces that contain more than four vertices. 

In this situation sense you are trying to only bevel the center of the model try using the inset tool. 

This can be accomplished by the "I" key while you have the desired faces selected. 

Here is a link for how to use the inset tool from the Blender documentation. 

https://www.blender.org/manual/modeling/meshes/editing/duplicating/inset.html